IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 3, MARCH 2000
477
Technical Notes and Correspondence_______________________________
A New Method for the Nonlinear Transformation of Means
and Covariances in Filters and Estimators
Simon Julier, Jeffrey Uhlmann, and Hugh F. Durrant-Whyte
Abstract—This paper describes a new approach for generalizing the
Kalman filter to nonlinear systems. A set of samples are used to parameterize the mean and covariance of a (not necessarily Gaussian) probability distribution. The method yields a filter that is more accurate than
an extended Kalman filter (EKF) and easier to implement than an EKF or
a Gauss second-order filter. Its effectiveness is demonstrated using an example.
z(k) = h[x(k); u(k); k] + w(k)
(1)
where x(k) is the state of the system at timestep k; u(k +1) is the input
vector, v(k) is the noise process caused by disturbances and modelling
errors, z(k) is the observation vector, and w(k) is additive measurement noise. It is assumed that the noise vectors v(k) and w(k) are zero
mean and
T
E [v(i)v (j )] = ij Q(i)
T
E [w(i)w (j )] = ij R(i);
T
E [v(i)w (j )] = 0:
8 i; j
Index Terms—Covariance matrices, estimation, filtering, missile detection and tracking, mobile robots, nonlinear filters, prediction methods.
^ (ijj )
The MMSE estimate of x(k) is the conditional mean. Let x
be the mean of x(i) conditioned on all of the observations up to and
including time j
I. INTRODUCTION
x^ (ijj ) = E [x(i)jZj ]
The problem of generalizing the Kalman filter paradigm for nonlinear applications is considered. This work is motivated by the wellknown limitations of the extended Kalman filter (EKF), which simply
linearizes all nonlinear models so that the traditional linear Kalman
filter can be applied. Although the EKF (in its many forms) is a widely
used filtering strategy [1], [2], over 30 years of experience with it has
led to a general consensus within the tracking and control community
that it is difficult to implement, difficult to tune, and only reliable for
systems that are almost linear on the time scale of the update intervals.
As is well known, the optimal solution to the nonlinear filtering
problem is infinite dimensional [3] and a large number of suboptimal
approaches have been developed [4], [5]. These methods can be broadly
classed as numerical Monte Carlo [6] methods or analytical approximations [7]–[9]. However, the application of these methods to high-dimensioned systems is rarely practical, and it is a testament to the conceptual simplicity of the EKF that it is still widely used.
In this paper, a new linear estimator [10]–[12] is developed. It yields
performance equivalent to the Kalman filter for linear systems, yet generalizes elegantly to nonlinear systems without the linearization steps
required by the EKF. We show analytically that the expected performance of the new approach is superior to that of the EKF and, in fact,
lies between those of the modified, truncated second-order filter [13]
and the Gaussian second-order filter [14]. However, the algorithm is not
restricted to Gaussian distributions. We demonstrate the performance
benefits in an example application, and we argue that the ease of implementation and more accurate estimation features of the new filter
recommend its use over the EKF in virtually all applications.
II. BACKGROUND
We seek the minimum-mean-squared error (MMSE) estimate of the
state vector of the nonlinear discrete time system
x(k + 1) = f [x(k); u(k); v(k); k]
where Zj = [z(1); 1 1 1 ; z(j )]: The covariance of this estimate is denoted P(ijj ):
The Kalman filter propagates the first two moments of the distribution of x(k) recursively and has a distinctive “predictor-corrector”
^ (k jk ); the filter first predicts what the
structure. Given an estimate x
future state of the system will be using the process model. Ideally, the
predicted quantities are
x^ (k + 1jk) = E [f [x(k); u(k); v(k); k]jZk ]
P(k + 1jk) = E [fx(k + 1) 0 x^ (k + 1jk)g
2 fx(k + 1) 0 x^(k + 1jk)gT jZk ]:
The expectations can be calculated only if the distribution of x(k)
conditioned on Zk is known. In general, the distribution cannot be described by a finite number of parameters and most practical systems
employ an approximation of some kind. It is conventionally assumed
that the distribution of x(k) is Gaussian at any time k: Two justifications are made. First, only the mean and covariance need to be maintained. Second, given just the first two moments the Gaussian distribution is the entropy maximizing or least informative distribution [15].
The estimate at time k + 1 is given through updating the prediction
by the linear update rule
x^ (k + 1jk + 1) = x^ (k + 1jk) + W(k + 1) (k + 1)
P(k + 1jk + 1) = P(k + 1jk) 0 W(k + 1)P (k + 1jk)
1 WT (k + 1)
^(k + 1jk )
(k + 1) = z(k + 1) 0 z
1
W(k + 1) = Px (k + 1jk)P0
(k + 1jk ):
~ (ijj ) =
The EKF exploits the fact that the error in the prediction, x
x(i) 0 x^ (ijj ); can be attained by expanding (1) as a Taylor Series about
^ (k jk ): Truncating this series at the first order yields the
the estimate x
approximate linear expression for the propagation of state errors as
Manuscript received November 16, 1994; revised December 1, 1995,
September 1, 1996, and June 1, 1998. Recommended by Associate Editor, V.
Solo.
S. Julier is with IDAK Industries, Jefferson City, MO USA (e-mail:
[email protected]).
J. Uhlmann is with the Robotics Research Group, Oxford University, U.K.
H. F. Durrant-Whyte is with the Department of Mechanical and Mechatronic
Engineering, Sydney, Australia (e-mail: [email protected]).
Publisher Item Identifier S 0018-9286(00)02145-0.
x~ (k + 1jk) rfx x~ (kjk) + rfv v(k)
where rfx is the Jacobian of f [1] with respect to x(k); and rfv is that
with respect to v(k): Using this approximation, the state prediction
equations are
x^ (k + 1jk) = f [^x(kjk); u(k); 0]
0018–9286/00$10.00 © 2000 IEEE
478
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 3, MARCH 2000
P(k + 1jk) = rfx P(kjk)rT fx + rfv Q(k + 1)rT fv :
n
(2)
This approximation is valid if the contributions of the truncated
higher order terms are negligible. However, in many practical situations (such as converting from polar to Cartesian coordinate systems
[17]), linearization introduces significant biases or errors. The EKF
can also be prohibitively difficult to implement because deriving
Jacobians can be cumbersome and time consuming (especially when
the system is complicated and of high order), and they must be
reevaluated at every prediction step of the filter.
This paper focuses on the central problem of predicting the mean and
covariance. We present an alternative algorithm that is more accurate,
convenient, and uses approximately the same number of calculations
as an EKF.
=
=
P(kjk):
x
P
x
X0 (kjk) = x^(kjk)
W0 = =(n + )
Xi (kjk) = x^(kjk) +
P jk)
(n + ) (k
Wi = 1=f2(n + )g
Xi+n (kjk) = x^(kjk) 0 (n + )P(kjk)
Wi+n = 1=f2(n + )g:
2 <; (
i
i
(3)
P j
P j
(n + ) (k k ))i is the ith row or column1 of the matrix
of (n + ) (k k), and Wi is the weight that is associated
square root
with the ith point.
Theorem 1: The set of samples chosen by (3) have the same sample
mean, covariance, and all higher odd-ordered central moments as the
distribution of (k): The matrix square root and κ affect the fourth and
higher order sample moments of the sigma points.
Proof: The matching of the mean, covariance, and all odd-ordered moments can be readily demonstrated. Because the points are
symmetrically distributed and chosen with equal weights about ; the
sample mean is obviously and all odd-ordered moments are zero. The
sample covariance is
x
P=
=
2n
i=0
n
i=1
P
x
x
2Wi (n + )
A P
P(kjk)
i
P(kjk)
P=A A
T
i
1If the matrix square root
of is of the form
, then the sigma
points are formed from the rows of : However, for a root of the form
, the columns of
are used.
AA
A
A
P=
T
i
2) The predicted mean is computed as
x^(k + 1jk) =
2n
i=0
Wi Xi (k + 1jk):
(4)
3) The predicted covariance is computed as
P(k + 1jk) =
2n
i=0
Wi fXi (k + 1jk) 0 x^ (k + 1jk)g
2 fXi (k + 1jk) 0 x^(k + 1jk)gT :
(5)
The mean and covariance are calculated using standard vector and
matrix operations, which means that the algorithm is suitable for any
choice of process model, and implementation is extremely convenient
because it is not necessary to evaluate the Jacobians, which are needed
in an EKF. The method has a further advantage: it yields more accurate
predictions than those determined through linearization.
Theorem 2: The prediction algorithm introduces errors in estimating the mean and covariance at the fourth and higher orders in the
Taylor series. These higher order terms are a function of κ and the
matrix square root used.
Proof: Appendix II shows that evaluating the mean and covariance correctly up to the mth order requires approximating the moments
of (k) up to the mth order. From Theorem 1, the approximation is correct up to the third order, and so errors are introduced in the fourth and
higher orders. These are a function of the matrix square root and κ.
Remark 3: This is the same order of accuracy as the second-order
Gaussian filter [14], but without the need to calculate Jacobians or Hessians.
provides an extra degree of freedom to “fine tune” the higher order
moments of the approximation, and can be used to reduce the overall
prediction errors. When (k) is assumed Gaussian, Appendix I shows
that a useful heuristic is to select n + = 3: If a different distribution
is assumed for (k), then a different choice of might be more appropriate.
Although can be positive or negative, Appendix III shows that
when is negative it is possible to calculate a nonpositive, semidefinite
(k + 1jk ): This problem is not uncommon for methods that approximate higher order moments or probability density distributions (such
as those described in [5], [4], and [8]). In this situation, it is possible to
use a modified form of the prediction algorithm. The mean is still calculated using (4), but the “covariance” is evaluated about X0 (k + 1jk):
Appendix III shows that the modified form ensures positive semi-definiteness, and, in the limit (n + ) ! 0; the prediction is the same as
that of the modified, truncated second-order filter [4].
x
x
x
P
Wi fXi (kjk) 0 x^ (kjk)gfXi (kjk) 0 x^ (kjk)gT
P(kjk)
Xi (k + 1jk) = f [Xi (kjk); u(k); k]:
x
P
i=1
i
Appendix I sketches the principle of the argument for the higher
moments, and a full proof can be found in [21].
Remark 1: The above properties hold for any choice of the matrix
square root. Efficient and stable methods, such as the Cholesky decomposition, should be used.
Remark 2: κ can be any number (positive or negative) providing
that (n + ) 6= 0:
Given the set of samples generated by (3), the prediction procedure
is as follows.
1) Each sigma point is instantiated through the process model to
yield a set of transformed samples
III. THE NEW FILTER
We use the intuition that it is easier to approximate a probability
distribution than it is to approximate an arbitrary nonlinear function
or transformation [18]. Following this intuition, we generate a set of
points whose sample mean and sample covariance are ^ (kjk) and
(k jk ), respectively. The nonlinear function is applied to each of
these points in turn to yield a transformed sample, and the predicted
mean and covariance are calculated from the transformed sample.
Although this superficially resembles a Monte Carlo method, the
samples are not drawn at random. Rather, the samples are deterministically chosen so that they capture specific information about the distribution. In general, this intuition can be applied to capture many kinds
of information about many types of distribution [19], [20]. In this paper,
we consider the special case of i) capturing the mean and covariance of
an ii) assumed Gaussian distribution.
The n-dimensional random variable (k) with mean ^ (kjk) and covariance (kjk) is approximated by 2n +1 weighted samples or sigma
points selected by the algorithm
P(kjk)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 3, MARCH 2000
479
Remark 4: The method can be generalized to include the effects of
process and observation noise by appending the noise vectors to the
state vector [11].
IV. EXAMPLE APPLICATION
This section applies and compares the performance of the new filter
against an EKF for a tracking problem that was presented in [14]. This
example was chosen because it has significant nonlinearities in the
process and observation models and has been analyzed extensively in
the literature.
We wish to estimate the position x1 (t); velocity x2 (t), and constant
ballistic coefficient x3 (t) of a body as it reenters the atmosphere at
a very high altitude at a very high velocity. Its motion is determined
by altitude- and velocity-dependent drag terms, and it is constrained to
fall vertically. The position of the body is measured at discrete points in
time using a radar capable of measuring range corrupted by Gaussian
measurement noise. The radar is at an altitude of H (100 000 ft), and
the horizontal range between the body and the radar, M; is (100 000
ft).
The continuous time dynamics of this system are
x_ 1 (t) =0x2 (t) + w1 (t)
x_ 2 (t) =0e0x
(t)
Fig. 1.
Absolute mean error position error.
(6)
2
x2 (t) x3 (t) + w2 (t)
x_ 3 (t) =w3 (t)
(7)
(8)
where w1 (t); w2 (t), and w3 (t) are zero-mean, uncorrelated noises
with covariances given by Q(t) and is a constant (5 2 1005 ) that
relates the air density with altitude. The range at time t; z (t); is
z (t) =
(M 2 + [x1 (t)
0 H ] ) + r(t);
2
(9)
where r(t) is the uncorrelated observation noise with covariance
4
2
= 10 ft . The measurements are made with a frequency of 1
Hz.
Tracking systems were implemented using the new filter and the
EKF. The nonlinearities of the process model and the high velocities
required the numerical integration of (6)–(8) to be carried out using
extremely small time steps. In accordance with [14], a fourth-order
Runge–Kutta scheme was employed with 64 steps between each observation. For the EKF, it was necessary to recalculate the Jacobian 64
times between each update. For the new filter, the trajectory of each
sigma point was calculated using the small time steps, but it was only
necessary to calculate the mean and covariance just before an observation was made. Because n = 3; was chosen to be zero in accordance
with the heuristic n + = 3:
The initial true state of the system is x(0) = [3 2 105 2 2 104 1003 ],
and the initial estimates and covariances of these states are
R(t)
x^ (0j0) = [3 2 105 2 2 104 3 2 1005 ]
6
P(0j0) =
10
0
0
0
4
0
2 10
0
0
10
6
:
Although the initial estimates of altitude and velocity are correct,
x^ 3 (0j0) is very bad. The body is assumed to be “heavy,” whereas in
reality it is “light.” The behavior of the two filters differs if the second
and higher order terms are significant. Because process noise can be
used to mask linearization errors, we adopt the practice from [14] and
do not introduce any process noise into the simulation—Q(k) = 0
for both filters.
In Fig. 1, we show the average magnitude of the state errors committed by each filter across a Monte Carlo simulation consisting of 50
Fig. 2. Absolute mean velocity error.
runs. At high altitude, the drag effects are minimal, and the body falls
approximately linearly. However, after about 10 s, drag becomes significant, motion becomes noticeably nonlinear, and the two filters differ
significantly. The velocity estimates are shown in Fig. 2 and indicate
large error spikes in both filters. These occur when the altitude of the
body is the same as that of the radar and range information provides
less data about body motion.The new filter recovers quickly, but the
EKF has a larger error spike and only slowly converges. Fig. 3 shows
the errors in estimating x3 (1): The error in the EKF estimate is biased
and is an order of magnitude larger than that for the new filter.
In Figs. 4 and 5, we show the errors in the position estimates made
by the EKF and the new filter and the associated estimates of the two
standard deviation bounds. These bounds are given by twice the square
root of the diagonals of the covariance matrix, and, if the filter is consistent, the state errors should lie within these bounds 95% of the time.
However, the error in the EKF drifts outside of these bounds after 30 s,
showing that it does not yield consistent estimates. However, the errors
in the new filter always lie well within the two standard deviations, implying that the new filter is consistent.
480
Fig. 3.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 3, MARCH 2000
Absolute mean error in x
:
Fig. 5.
New filter mean position error.
This algorithm has found a number of applications in high-order,
nonlinear coupled systems, including navigation systems for
high-speed road [21], [16] vehicles, public transportation systems
[22], and underwater vehicles [23]. Square root filters can be formulated by discrete sets of points (as demonstrated in [24]), and
iterated filters can be constructed using the predictions [25]. The
algorithm has been extended to capture the first four moments of a
Gaussian distribution [12] and the first three moments of an arbitrary
distribution [20].
Given its performance and implementation advantages, we conclude
that the new filter should be preferred over the EKF in virtually all
nonlinear estimation applications.
APPENDIX I
HIGHER MOMENTS OF THE APPROXIMATION
Fig. 4.
EKF mean position error.
Therefore, we conclude that in this example the new filter has substantial advantages over the EKF both in implementation and performance.
V. CONCLUSIONS
Motivated by the deficiencies of the EKF, we have examined a completely new approach for applying linear estimation theory to nonlinear systems. Rather than approximate the Taylor series to an arbitrary order, we approximate the first three moments of the prior distribution accurately using a set of samples. The algorithm predicts the
mean and covariance accurately up to the third order and, because the
higher order terms in the series are not truncated, it is possible to reduce the errors in the higher order terms as well. We have provided
empirical evidence that supports the theoretical conclusion and have
demonstrated that the new filter is far easier to implement because it
does not involve any linearization steps, eliminating the derivation and
evaluation of Jacobian matrices.
This appendix analyzes the properties of the higher moments of the
sigma point selection scheme of (3). The selection process consists of
three stages. First, a set of sigma points are drawn to approximate an
n-dimensional standard Gaussian with mean 0 and covariance I: A
linear transformation [a matrix square root P(kjk )] is applied to each
point so that the transformed sample still has mean 0, but the covari^ (k jk ) is added to each transformed sigma
ance is P(kjk): Finally, x
point to ensure the correct sample mean. affects the first stage of the
approximation, and the matrix square root affects the second.
The sigma points, which are assumed approximate the standard
Gaussian, lie on the coordinate axes.2 The orthogonality and the symmetry of these points means that the only nonzero sample moments
are those that are an even order power of a single coefficient. The
(k01)
2k th-order moment for any coefficient is (n + )
: Therefore,
the higher order moments scale geometrically by a factor determined
by . However, the moments for the standard Gaussian are different
from those of the sigma points, which approximate it. Because the
covariance matrix is I, the different components are independent of
one another and
E [x1
2In
I:
2 111 2
xn
n
]=
i=1
E [xi
]
effect, these points are drawn from the orthogonal matrix square root of
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 3, MARCH 2000
481
where the ith component in the moment is raised to the power i : It
can be shown that [13]
E [xi
] =
i is odd:
;
0
1
2 2 1 1 1 2 i 0 ; i is even:
3
1)
(
Two differences exist in the approximation by the sigma points.
First, the “scaling” of nonzero moments are different. Second, a
number of moment terms are “missed out.” These differences become
more marked as the order increases, but, as explained in Appendix II,
they only affect the higher order terms of the Taylor Series of the
process model/nonlinear function. Assuming that the contribution of
terms diminishes as the order increases, the emphasis is to minimize
the errors in the lowest order terms. Because E [x4i ] = 3 for the
Gaussian distribution, choosing (n + ) = 3 minimizes the difference
between the moments of the standard Gaussian and the sigma points
up to the fourth order.
Because the fourth and higher order moments are not precisely
matched, the choice of matrix square root affects the errors in the
higher order terms by adjusting the way in which the errors are
distributed among the different states. However, in general, this information cannot be exploited because it would require knowledge of the
higher order derivatives of the process model equation. In the absence
of this information, the choice of matrix square root is governed by
other issues, such as numerical stability or computational cost.
=
y
y fx
x x x
P
x
y
x
x
x
x
x
y fx x
2
3
4
f x D1xf D1xf D1xf D1xf 1 1 1 (10)
where the D1x f operator evaluates the total differential of f 1 when
perturbed around a nominal value x by x: The ith term in the Taylor
series for f 1 is given by
i
n
D1i xf
(11)
xj @ f x
=
[
=
[
+ ~]
] +
+
+
2!
+
3!
4!
[ ]
[ ]
=
1
i! j =1
~
@xj
[
y
y
=
=
x
E [f [x + x~ ]]
fx
[
] +
E
2
3
4
D1xf D1xf D1xf D1xf 1 1 1
+
2!
+
+
3!
=
D1i xf
i!
=
1
i!
E
n
j =1
xj @x@ j
~
i
fx
[
]
+ ~]
[
=
111
(13)
+ ~ ]]
[ [
+
+
2!
2!
+
+
3!
4!
(14)
+
4!
with substitutions from (10) and (12). The true covariance is found
by postmultiplying the state error by the transpose of itself and taking
expectations. Exploiting the fact that ~ is symmetric, the odd terms all
evaluate to zero and the covariance is given by
x
Pyy rfxPxxrfxT
3 T D12 x f D12 x f T
E D1x f D1x f
(12)
(
+
+
)
3!
(
+
2
2
D13 xf D1xf T 0 E D12 xf
(
)
2!
E
)
3!
2
D12 xf
T
2
+
111
(15)
D fD f
fP
f
where E [ 1x ( 1x )T ] = r x xx r xT has been used. Comparing
this with (2), it can be seen that the EKF truncates this series after the
first term. The new filter does not truncate the series at any arbitrary
order, and, applying the analysis from above, it is correct up to the
second order with errors in the fourth and higher order terms. Therefore, both the EKF and the new filter predict the covariance correctly
up to the third order.
APPENDIX III
THE MODIFIED FORM OF THE ALGORITHM
When is negative, it is possible that the predicted covariance will
not be positive semidefinite. This can be demonstrated by taking the
limit of (15) as (n + ) ! 0: As shown in Appendix I, the fourth and
higher order moments tend to zero and
D12 xf
T
(n+)!0 Pyy rfx Pxx rfx 0 E
lim
=
2
E
D12 xf
2
T
:
It can be seen that a fourth-order, positive-semidefinite matrix is subtracted. However, this term originates from the outer product of two
second-order expected terms and does not scale with .
The modified form of the algorithm evaluates the covariance about
the projected mean. Taking the sigma points for are Xi and those for
are Yi ; the modified form for calculating yy is3
y
PMOD
yy
where the ith term in the series is given by
E
4!
+
+
y y y y
y0y f x x 0E f x x
2
3
4
D1xf D1xf D1xf D1xf
2
4
0 E D1x f D1x f 1 1 1
]
where x
~j is the j th component of ~ : Therefore, the ith term in the series
is an ith-order polynomial in the coefficients of ~ ; whose coefficients
are given by derivatives of [1]:
is the expected value of (10)
f
2
P
x=X
x
1
x
+
~
i!
f
@i
i
@x 01 @x
m111112
=
In this appendix we provide a justification for approximating a probability distribution by a set of samples that match its moments. We show
that an approximation is only correct to the mth order if its moments
up to that order are correctly approximated.
Consider a Gaussian-distributred random variable with mean
and covariance xx : We wish to calculate the mean and covariance
yy of the random variable , which is related to through the nonlinear (analytic) function = [ ]:
Noting that can be written as = + ~ , where ~ is a zero-mean
Gaussian random variable with covariance xx ; the nonlinear transformation can be expanded as a Taylor Series about
P
i!
+
and mc c 111c is the ith-order moment mc c 111c = E [x
~c x
~c 1 1 1 x
~c ]:
The ith term is a function of the ith central moments of the distribution of ~ : Therefore, for an approximation to be accurate up to the ith
order, it must be able to approximate the moments correctly up to the
ith order. The EKF truncates this series after the first term, and so its
error in predicting the mean is in the second and higher orders. The new
filter matches the mean and covariance correctly, and so it is correct up
to the second order. Because the new filter does not truncate the filter
at any order, it can be expected that the errors in the fourth and higher
order terms are smaller than those committed by the EKF.
The covariance is given by yy = E [f 0 gf 0 gT ]: Now
APPENDIX II
MOMENT APPROXIMATION AND PERFORMANCE
P
i
m111111 @ fi
@x1
1
2n
=
i=1
P
x
Wi [Yi 0 Y0 ][Yi 0 Y0 ]T :
3This is equivalent to calculating the mean and covariance using (4) and (5)
and adding a term fy
y
:
= Y gf 0 Y g
482
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 3, MARCH 2000
Positive semidefiniteness is guaranteed by the fact that the covariance matrix is evaluated as the sum of outer products of vectors. It can
be shown that the calculated covariance is
Pyy
=r
fxPxx fxT
D1xf D31xf T D12 xf D12 xf T
r
(
+ E
)
+
D13 xf D1xf T
(
(
+
3!
)
)
2 2 2!
+ 111
3!
which introduces errors in the fourth and higher orders.
The modified form has the advantage that, irrespective of the choice
of the matrix square root
D12 xf
y
f
x
(n+)!0
fxPxx fxT
(n+)!0 Pyy
lim
lim
=
[
] + E
=r
r
2!
:
These are the values calculated by the modified, truncated secondorder filter [4], but without the need to evaluate Jacobians or Hessians.4
ACKNOWLEDGMENT
The authors would like to thank R. Bellaire at Georgia Tech for insightful comments on an early draft of this paper and the referees for
their many constructive remarks.
REFERENCES
[1] J. K. Uhlmann, “Algorithms for multiple target tracking,” Am. Sci., vol.
80, no. 2, pp. 128–141, 1992.
[2] H. W. Sorenson, Kalman Filtering: Theory and Application. New
York: IEEE Press, 1985.
[3] H. J. Kushner, “Dynamical equations for optimum non-linear filtering,”
J. Differential Equations, vol. 3, pp. 179–190, 1967.
[4] P. S. Maybeck, Stochastic Models, Estimation, and Control, P. S. Maybeck, Ed. New York: Academic, 1982, vol. 2.
[5] A. H. Jazwinski, Stochastic Processes and Filtering Theory. New
York: Academic, 1970.
[6] N. J. Gordon, D. J. Salmond, and A. F. M. Smith, “Novel approach
to nonlinear/non-Gaussian Bayesian state estimation,” Proc. Inst. Elect.
Eng. F, vol. 140, no. 2, pp. 107–113, Apr. 1993.
[7] H. J. Kushner, “Approximations to optimal nonlinear filters,” IEEE
Trans. Automat. Contr., vol. AC-12, pp. 546–556, Oct. 1967.
[8] H. W. Sorenson and A. R. Stubberud, “Non-linear filtering by approximation of the a posteriori density,” Int. J. Contr., vol. 8, no. 1, pp. 33–51,
1968.
[9] F. E. Daum, “New exact nonlinear filters,” in Bayesian Analysis of
Time Series and Dynamic Models, J. C. Spall, Ed. New York: Marcel
Dekker, 1988, pp. 199–226.
[10] S. J. Julier, J. K. Uhlmann, and H. F. Durrant-Whyte, “A new approach
for filtering nonlinear systems,” in Proc. Am. Contr. Conf., Seattle, WA,
1995, pp. 1628–1632.
[11] S. J. Julier and J. K. Uhlmann. (1994, Aug.) A general method for approximating nonlinear transformations of probability distributions. [Online]. Available: http://www.robots.ox.ac.uk/~siju
[12]
, “A new extension of the Kalman filter to nonlinear systems,”
in Proc. AeroSense: 11th Int. Symp. Aerosp./Defense Sensing, Simulat.
Contr., Orlando, FL, 1997.
4This can be contrasted with an alternative approach of the initial intuition,
which was explored in [26]. Under that scheme, no copies of the previously
estimated mean are included in the sample set and the sigma points are scaled
using a parameter . In the limit, as tends to infinity, this algorithm predicts
the same mean and covariance as the EKF. However, when 1; this method
estimates the same mean and covariance as the new filter, but with = 0:
=
[13] P. S. Maybeck, Stochastic Models, Estimation, and Control. New
York: Acdemic, 1979, vol. 1.
[14] M. Athans, R. P. Wishner, and A. Bertolini, “Suboptimal state estimation for continuous-time nonlinear systems from discrete noisy measurements,” IEEE Trans. Automat. Contr., vol. 13, pp. 504–518, Oct. 1968.
[15] D. E. Catlin, “Estimation, control and the discrete Kalman filter,” in
Applied Mathematical Sciences 71. New York: Springer-Verlag, 1989,
p. 84.
[16] S. Clark, “Autonomous land vehicle navigation using millimetre wave
radar,” Ph.D. dissertation, University of Sydney, Australia, 1999.
[17] D. Lerro and Y. K. Bar-Shalom, “Tracking with debiased consistent converted measurements vs. EKF,” IEEE Trans. Aerosp. Electron. Syst., vol.
29, pp. 1015–1022, July 1993.
[18] J. K. Uhlmann, “Simultaneous map building and localization for real
time applications,” Univ. Oxford, U.K., Tech. Rep., 1994.
[19] S. J. Julier and J. K. Uhlmann, “A consistent, debiased method for
converting between polar and Cartesian coordinate systems,” in Proc.
AeroSense: 11th Int. Symp. Aerosp./Defense Sensing, Simulat. Contr.,
Orlando, FL, 1997.
[20] S. J. Julier, “A skewed approach to filtering,” in Proc. AeroSense: 12th
Int. Symp. Aerosp./Defense Sensing, Simulat. Contr., Orlando, FL, 1998.
, “Comprehensive process models for high-speed navigation,”
[21]
Ph.D. dissertation, Univ. Oxford, U.K., 1997.
[22] A. Montobbio, “Sperimentazione ed affinamento di un localizzatore,”
B.S. thesis, Politechnico di Torino, Italy, 1998.
[23] R. Smith, “Navigation of an underwater remote operated vehicle,” Univ.
Oxford, Tech. Rep., 1995.
[24] T. S. Schei, “A finite difference approach to linearization in nonlinear
estimation algorithms,” in Proc. Am. Contr. Conf., vol. 1, Seattle, WA,
1995, pp. 114–118.
[25] R. L. Bellaire, E. W. Kamen, and S. M. Zabin, “A new nonlinear iterated
filter with applications to target tracking,” in Proc. AeroSense: 8th Int.
Symp. Aerosp./Defense Sensing, Simulat. Contr., vol. 2561, Orlando, FL,
1995, pp. 240–251.
[26] B. M. Quine, J. K. Uhlmann, and H. F. Durrant-Whyte, “Implicit Jacobians for linearized state estimation in nonlinear systems,” in Proc.
Amer. Contr. Conf., Seattle, WA, 1996, pp. 1645–1646.
An
Boundary Layer in Sliding Mode for
Sampled-Data Systems
Wu-Chung Su, Sergey V. Drakunov, and Ümit Özgüner
Abstract—The use of a discontinuous control law (typically, sign functions) in a sampled-data system will bring about chattering phenomenon
in the vicinity of the sliding manifold, leading to a boundary layer with
is the sampling period. However, by proper
thickness ( ) where
consideration of the sampling phenomenon in the discrete-time sliding
mode control design, the thickness of the boundary layer can be reduced
to ( ) In contrast to discontinuous control for continuous-time VSS,
the discrete-time sliding mode control need not be of switching type.
Index Terms—Boundary layer, discrete-time sliding mode, sampled-data
systems.
Manuscript received April 3, 1998; revised August 31, 1998, and February
28, 1999. Recommended by Associate Editor, S. Hara. This work was
supported by the National Science Council of Taiwan, under Project NSC
89-2213-E-005-026.
W.-C. Su is with the Department of Electrical Engineering, National ChungHsing University, Taichung, Taiwan (e-mail: [email protected]).
S. V. Drakunov is with the Department of Electrical and Computer Engineering, Tulane University, New Orleans, LA 70118 USA.
Ü. Özgüner is with the Department of Electrical Engineering, The Ohio State
University, Columbus, OH 43210 USA.
Publisher Item Identifier S 0018-9286(00)02146-2.
0018–9286/00$10.00 © 2000 IEEE